Knowledge that Transforms

The general procedure followed in the present paper is to show that, although the series generated by the Theil-Wage model [Theil, H., S. Wage. 1964. Some observations on adaptive forecasting. Management Sci. 10.] is nonstationary, there exists a simple transform of the series, in this case the second difference, which is stationary. This observation permits the Wiener-Hopf theory for stationary series to be applied to the transformed series. It is then shown that the results obtained by Theil and Wage are simply related to the optimal constant-parameter, linear predictors of the transformed series and thus that the adaptive forecasts are optimal in a rather wide sense. We believe, therefore, that the results of this paper illustrate a general approach to the prediction of non-stationary time series, and these are, after all, the type mainly encountered in economic or management problems. Thus the paper may have a somewhat wider significance than its title or primary purpose might suggest.

The problem of finding an optimum, no-risk strategy for placing simultaneous win and place pari-mutuel bets is formulated in terms of a linear programming problem. For a basic feasible solution (i.e., a profitable schedule of bets) to exist, it is necessary for the win and place pools to weight the contestants differently. If the stake holder removes a fraction of the pool, then a solution is not guaranteed, but will exist if this difference in win-place weighting is sufficient. A simple example is presented.

The organizers of this meeting have asked me to summarize the discussions of the last three days and to draw some conclusions. These will have to be provisional and personal conclusions only, formulated in all modesty, since I am not an expert in several of the more sophisticated methods discussed here. While the term “operational research” has not, as far as I know, a very precise meaning, we may nevertheless characterize it by: (1) treating complicated problems with a large number of variables; (2) using various modern mathematical and logistic methods; and (3) being directed toward a goal of action. Its success has been proved already especially solving a number of problems in manufacturing industries, transportation and energy, fields for which usually precise and abundant data are available which make it worthwhile applying sophisticated methods. When we ask whether this complex of methods can also be applied to development problems, it seems appropriate to summarize also what seems to be characteristic for these problems.

An individual uses up a certain resource at a constant rate, but from time to time has a chance to gamble some of the resource in order to gain more. The expectation of each gamble is zero, i.e., the gambles are fair. This means that eventually the resource will run out. However, the form of the risk can be varied arbitrarily. The question is what form of risk to choose, when the object is maximize the expectation of the utility, u(T), of having the resources run out at time T. This problem is solved for a more or less arbitrary function u(T). Various possible interpretations of such a function u(T) are discussed briefly. The model is mainly intended to provide a basis for empirical studies of individual decision making, where its complete mathematical tractibility is convenient.

The paper evaluates and compares classes of multi-item inventory problems, where joint order of several items may save a part of the setup cost. A cost ratio and simple decision rule are determined for joint versus individual orders in specified cases. The comparisons call for the necessity of a new policy for reorder point-triggered random output multi-item systems. This policy, the “random joint order policy,” operates through the determination of a reorder range within which several items can be ordered. The existence of an optimum reorder range is proved, and a computational technique is demonstrated with the help of a machine-interference type queueing model. Under favorable conditions the results exhibited on the diagram of optimum reorder ranges are generally applicable. The random joint order policy model is especially suitable for computer controlled inventory systems.

The selection of appropriate product strategies by firms must account for the expected long-run returns from the market a firm contemplates entering, relative to the expected long-run returns from the market it is currently serving. These expected returns must also reflect the capabilities a firm has to implement this strategy and the resulting competitive situations it will face. The mathematical techniques of Markov processes and dynamic programming are used in this paper to develop an analytic framework which will select optimal product strategies for a firm that are consistent with its capabilities, that will meet its competitive situations, and that will maximize expected returns in the long run. This model is applied to a selected set of Michigan machine tool firms. The results indicate the general importance of total firm capabilities and the greater importance of management capability over the supporting capabilities in the selection of optimal product strategies.

This paper describes a simple calculation for the assignment and transportation problems which is “dual to” the well-known Hungarian Method. While the Hungarian is a dual method, this method is primal and so gives a feasible assignment at each stage of the calculation. Bounds on the number of steps required for the assignment and transportation problems are given. They are the same as the best bounds known for the Hungarian Method.

A general model has been developed for a class of business games to represent the activities of individual firms in a hypothetical oligopolistic industry. Although hypothetical, the games allow the players to transfer mentally to the real business world. The players who act as competitors build and operate their individual plants with specified technology to produce a single homogeneous product. The product is sold competitively to households, and labor supplied by the households is employed in the industry through competitive bidding of the competitors. Thus, there are two competitive markets established, with the firms competing for a limited supply of labor on the one hand, and competing for a limited sales fund on the other hand. Managerial decisions for each firm include labor employment offerings and wage rate, product sales offerings and product prices, production and inventory, plant expansion, money borrowing, sales credit, and dividends paid to stockholders. In some cases there is a market for “efficiency units” used to improve technology. The competitors may select wage rates and product prices freely, since there are no internal relationships in the model. Aggressive competition for labor results in increasing wage rates and aggressive competition for the sales fund results in low product profit. Various regulations are introduced concerning monopoly, taxation, credit, and bankruptcy. The general results of this study show that managerial behavior in an oligopoly exhibits varying strength of agressive competition producing the usual business cycle, and further that competitors will act to “stabilize” the industry by self-regulation of speculative activities. An examination of the technological input to these business games shows that competitors will invest funds to improve efficiency only when their marginal utility of money is well below that represented by the fixed interest rate.

A systematic optimization procedure is developed for the design of a plant which is subject to uncertain, seasonal production. Based on a generalized seasonal production model, the present value of the cash flows associated with production by means of a combination of alternates is expressed in terms of the unknown capacities. On the basis of the derived solution, a systematic procedure is presented which yields optimum designs without reference to the involved mathematics. Finally, the more important assumptions are removed and suggestions are given with respect to the modifications that are required in order to extend the method to more complex problems.

Matrix methods for various “bill of materials” computations have been described in papers by Vazsonyi [Vazsonyi, A. 1954. The use of mathematics in production and inventory control. Management Sci. 1 (1, October) 70–85; Vazsonyi, A. 1958. Scientific Programming in Business and Industry. Wiley, 429–452.], Vazsonyi and Larson [Vazsonyi, A., H. T. Larson. 1955. Data-processor requirements in production and inventory control. Proceedings of the Western Joint Computer Conference, 48–61.], and Giffler [Giffler, B. Mathematical solution of production planning and scheduling problems. IBM Technical Report 09.01028.026.]. Wide-spread use of these ideas has been hampered by the large amount of computation that the basic “total-requirements” matrix T involves. In this paper, a new and efficient iterative method for engineering changes of T is discussed, and this method is applied to the original generation of T, and to the inversion of certain matrices.